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The Bloch Constant of Bounded Analytic Functions
Author(s) -
Colonna Flavia
Publication year - 1987
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-36.1.95
Subject(s) - unit disk , bounded function , analytic function , conformal map , automorphism , blaschke product , product (mathematics) , constant (computer programming) , mathematics , unit (ring theory) , sequence (biology) , combinatorics , function (biology) , entire function , mathematical analysis , geometry , computer science , programming language , mathematics education , evolutionary biology , biology , genetics
In this work we solve the extremal problem of characterizing all bounded analytic functions f : Δ → C (where Δ is the open unit disk) for which the Bloch constant β f = sup{(1−|z| 2 )| f ′(z)|: z ε Δ is a bound. Normalizing, we study f : Δ → Δ with β f = 1. We show that these are precisely the conformal automorphisms of Δ together with those functions whose zeros form an infinite sequence ( z n ) nεN such thatlim sup n → ∞| g ( z n ) | Π k ≠ nz n − z k1 − z n z k= 1 ,where g is the non‐vanishing function such that f/g is a Blaschke product. In particular, non‐vanishing inner functions, finite Blaschke products, and outer functions for the class H ∞ (Δ) with image contained in Δ are not extremal functions.